Mistake

I've just (mistakenly) did an update to this entry. No changes have been made at all. The update I meant to do was that of the Portuguese version, which I have just forked off the English one. --Doshell

Redirect issues

"Real numbers" has just been redirected to "Real number". Fine with me, but presently "Rational number" redirects to "Rational number".

What do we want to do, Wikipedians?

Thanks for noticing that "rational number" has the same problem--it's fixed now. The singular form is preferred for simple nouns like this to make linking easier. For example, "...probability can be expressed as a real number in the interval [0,1]..." --LDC

In my opinion, the end page of redirections should be Real numbers. This causes no problems for linking to Real number as in LDC's example, as Real number would be diverted to Real numbers. The reason for this preference (and the same for Complex numbers, etc.) is that the field of real numbers is constructed as a whole, and a single real number may only be defined by its relation to other numbers. This is different to articles like Group (mathematics), where the axioms that distinguish a group make no external references, and working within a single group is not uncommon. Elroch 13:12, 22 February 2006 (UTC)

WP standard is to use singular titles whenever it makes sense to do so. A single real number makes sense by itself (for example, in terms of the information it codes), so the article should be at real number. There are a few special cases where only a plural title makes sense, like orthogonal polynomials or indiscernibles, but that isn't the case here. --Trovatore 14:20, 22 February 2006 (UTC)

Using the singular form also makes it easier to link using either the singular and plural forms. If the article has a title in the singular then the plural can be obtained just by adding an s on the end [[real number]]s giving real numbers. Using the reverse makes it harder to do a link to the singular form you need [[real numbers|real number]]. --Salix alba (talk) 14:25, 22 February 2006 (UTC)

I am happy to join the consensus for using singulars for article titles. However, the issue of linking is unimportant, as links to "real number" would work as intended if "real number" was diverted to "real numbers", in the same way as links to "real numbers" work fine at present. Elroch 15:28, 22 February 2006 (UTC)

Cauchy sequences

If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing). When applied to the rational numbers, it gives the following useful construction of the real numbers:

It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead.

Comment moved from the main page

The following comment was moved from the main page:

RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it.

That's the cohomology dimension; I don't see why it's better than any other.

Question

Tarquin, do you plan to add to the symbols for subsets of the real line? Otherwise, I'll add to it, including the uses of the symbols. Right now, it's not very clear. — Toby 07:16 Aug 3, 2002 (PDT)

Local field

As I understand it, local field means a field complete with respect to a discrete valuation. Are the reals and complexes really local fields? --alodyne

Oh and by the way: at least one change certainly must be made in that statement since the p-adic numbers are local fields of characteristic 0 by all the various definitions I've seen. Any dispute? --alodyne

No. The Encyclopedic Dictionary of Mathematics defines a local field as a field that's complete with respect to a discrete valuation and such that it's residue field is finite. They mention that the the reals and complex numbers are also sometimes considered as local fields, but they explicitly exclude them. AxelBoldt 19:14 Nov 12, 2002 (UTC)

To tie up this article with the entries on Model Theory, can someone tell me whether or not there is a first-order theory model for the real numbers? A maths teacher friend of mine told me that there is, but this would imply, by Lowenhein-Skolem theorems that the reals are denumerable, or have a countable model, which which seem to be inconsistent with Cantor's diagonal proof of uncountability of the Reals. --B. Smith.

No, it would not imply that. It would imply only that there is a countable field satisfying all of the same first-order sentences that the real numbers satisfy. The Loewenheim-Skolem theorem does imply the existence of such a model. If the model and the language are elaborate enough, one could write a first-order sentence saying the reals are not countable. It would be true within the countable model. That means that although there would be a sequence containing every member of the model, such a sequence could not itself belong to the model; there could be no enumeration within the model. Michael Hardy 00:12 Apr 11, 2003 (UTC)

"The term 'real number' is a retronym coined in response to 'imaginary number'." That hardly seems likely, since extension to the reals should always have seemed more urgent than extension to complex, and the reals need a name as soon as you consider them.

IMO, real is intended to contrast with rational, and the motivation for "real" is that the rationals are an unrealistically limited set. That follows from the mind necessarily regarding all integer-legged right triangles as precise models of something from reality, if anything in math beyond counting is such a model; from irrational hypotenuses, it follows that rationals aren't quite the full set of numbers needed to correspond to reality, so the choice of the term "real number" reflects the hypothesis that there are no problems in the real world that the reals don't suffice to express the answer to. (In contrast, wanting imaginaries and quaternions takes some sophisticated mathematical abstraction.)

In fact, the best evidence that "real" led to "imaginary" (not the other way around) is that "real" is a better fit to the reals than "imaginary" is to the imaginaries: IIRC how one of my physics profs put it, it's not so much a matter of complex numbers having an imaginary part as that they have *two* parts, both real numbers representing something in the physical world, and it's just that there are systems like electromagnetic fields whose behavior looks simpler if you do the bookkeeping by labeling the electric field as the first component of a complex number and the magnetic as the second, and hiding the fact that the "multiplication" you do is, according to the rules of complex arithmetic, more complicated than the multiplication table you learned in grade school.

So where's this "retronym" thing coming from; is it more than someone's conjecture? --Jerzy 06:41, 30 Sep 2003 (UTC)

I don't think there was a monolithic nomenclature that all mathematicians subscribed to at the time when negative numbers, imaginary numbers, and real numbers were being investigated and systematized (15th century through 19 century-ish). History is messy like that. It is well known that negative numbers (first) and imaginary numbers (later) inspired censure (and sometimes even disgust) in the European mathematicians of the day, so the retronym thing is at least plausible, e.g. "The square root of -1? That's not a real number, it's some kind of imaginary thing!" I'd like to see a source too though. -- Cyan 07:06, 30 Sep 2003 (UTC)

Just for information, the first reference to imaginary numbers in the Oxford English Dictionary is to Descartes (in French) in 1637, and the first reference in English is in 1706: W. JONES Syn. Palmar. Matheseos 127 The Original Components or Roots of all Equations, may be either Affirmative, Negative, Mix'd, or Imaginary. In contrast, the first reference to a real number that they give is in a 1727 encyclopedia: CHAMBERS Cycl. s.v. Root, If the value of x be positive, i.e. if x be a positive quantity,..the root [of an equation] is called a real or true root. (These are actually citations for the use of real or imaginary as adjectives with the modern meaning, so they weren't looking specifically for the phrase real number, which they don't cite until the 1910 Encyc. Brit.) Although I'm not sure the OED is as good about backdating mathematical terms as they are in general, this gives some indication at least. Anyway, since a rigorous definition and theory of real numbers wasn't really developed until the 19th century, it's hard to call it a retronym (what people had looked at in the past were real numbers but didn't define them; e.g. if you look at just the roots of polynomials, those are the algebraic numbers which are a countable subset of the reals). Steven G. Johnson 02:02, 12 Oct 2003 (UTC)

The article begins with:

The real numbers are practically any numbers that can be expressed.

I have to say I find this unsatisfying, especially since most (i.e. all but a countable subset of) real numbers arguably cannot be specifically expressed. That is, if you take the meaning of expressed in a natural way: i.e. to be uniquely defined by a finite-length description (such as 4.73, √2, sin(1), ...). And only a strict subset of these expressible numbers are computable (i.e. to an arbitrary precision in a finite time). Steven G. Johnson 01:34, 12 Oct 2003 (UTC)

It's impossible to explicitly specify a non-recursive number;

That depends on what "explicitly specify" means. Consider, for example, Chaitin's constant. This is non-recursive, but in some sense can be specified. Josh Cherry 23:31, 19 Oct 2003 (UTC)

And what is this Russian school that assumes all numbers are recursive? This sounds like a provably false statement to me, unless something nonobvious is meant by "recursive algorithm". It's provably impossible to specify every real numbers uniquely with a recursive algorithm in the computer-science sense, i.e. a finite-size finite-state machine, or equivalently a finite-length program in any Turing-complete computer language. (There are only a countable number of finite-length computer programs.) (One can even explicitly define a real number, whose digits are e.g. based on the halting problem, that is uniquely specified by a finite-length description but which is not computable in finite time by any program.) Steven G. Johnson 20:00, 21 Oct 2003 (UTC)

Certainly one can explicitly specify particular non-computable numbers. I haven't read everything above, so I'm not certain which "Russian school" is referred to, but it's probably something about constructivism, which is a philosophy that holds that an existence proof is not valid unless it "constructs" the object whose existence is to be proved. For example, if you were to deduce a contradiction from the proposition that every even number greater than 2 is a sum of two primes, that would not be taken by constructivists to be a proof of the existence of a counterexample. Michael Hardy 00:03, 22 Oct 2003 (UTC)

However, most (all but a countable subset of) real numbers cannot be uniquely specified by a finite description of any sort (whether by a computer algorithm or otherwise). The current Wikipedia statement about the Russian school needs clarification (or deletion), because on its face it seems to imply the contrary. Steven G. Johnson 04:11, 22 Oct 2003 (UTC)

Quantity Box

This has been removed several times by one person. I am among those who sees it as an enhancement of the article and the "explanations" for its removal wrong -- unless the word is just "silly". It's time to stop deleting and start defending the preceding deletions on this page: convince someone if you're so sure. --Jerzy(t) 01:56, 2004 Mar 28 (UTC)

The box is not silly but unnecessary and distracting. The box is already too long (you have to scroll to see the whole list and it has a potential to get longer certainly. This kind of the box is only useful and necessary in cases like 1) the article is separated into several pages because the article would be too long if those were in one page. 2) The article is about one entity among several ones. There is no much relationship between real and natural numbers except they are numbers. You want me to convince you then can I also ask you to convince me first? To my knowledge, there was no consensus about having such a box. In other words, I am just reverting the article to what it used to be and see if we can have concensus or something. -- Taku 03:04, Mar 28, 2004 (UTC)

Negatives are that recent?

From the History section: "Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians."

I find it hard to believe that negative numbers were just being accepted in Newton's time. I'm no history geek, but I'd like to get that confirmed.

Same here. Newton obviously used negative numbers, and what about Napier? I had thought that negative numbers were introduced by medieval Italian accountants. Michael Hardy 20:59, 6 Dec 2004 (UTC)

A serious mistake here. Chinese mathematicians (indeed they maybe more "accountants" than "mathematicians") developed similar concept long time ago. I believe that negative numbers have been reinvented many times, as surplus and deficiency are fundamental concepts in commerce. -wshun 11:45, 10 Dec 2004 (UTC)

Same seccion says that euler descarted solutions to equations as unrealistics. I don't know which equation and context, but it doesn't implies in anyway that he thought negative numbers as unrealistic!!!

Unless someone posts a *very* good source on that I remove it because it's ridiculous. Apart from the problem that the general level of European mathematics in the late 18th century makes it seem unrealistic that they didn't understand a concept so fundamental as negative numbers (Especially as Euler's work on complex numbers makes no sense without the concept of negative ones).

Also look here: http://www.uni-essen.de/didmath/texte/jahnke/hnj_pdf/euler01.pdf -- Even if you don't speak German a simple look at the numbers should make clear that Euler used the - as an inherent property of the number instead of just as operator. (Not to mention that apparently west of India you could spend as much money as you wanted once you were broke because noone could grasp the concept of debt; I also wonder how Europeans could come up with imaginary numbers before negative numbers were used in Europe...)

I assume whoever came up with that claim took valid concerns over solutions out of context (e.g. if I use Pythagoras to get the third side of a triangle, both positive and negative are solutions to that equation, but the negative doesn't make sense).

So unless someone comes up with a paragraph that's sourced with some credible sources, I'd say leave it out. 82.135.81.88 23:04, 6 May 2007 (UTC)

I'm surprised that remark sat there for nearly two years. Someone wrote "they were not generally accepted in Europe until...". Then on May 13th, 2005, an anonymous user changed it to ""they were not used in Europe until...". That was a very irresponsible edit. Michael Hardy 02:51, 7 May 2007 (UTC)

The history sections of many articles have an anti-European bias. They cite the first examples (sometimes ambiguous ones) of something being used outside of Europe and then contrast it with the last European doubter (e.g. Diophantus writes "a number to be subtracted, multiplied by a number to be subtracted, gives a number to be added." That doesn't necessarily mean that he "gets" negative numbers as a concept but the colored rods of the Chinese aren't any better -- note that I have read neither, I'm just going by wikipedia entries, but that means either the conclusions on wp are wrong or they present insufficient evidence to support their conclusions =).

There certainly is a pro-European bias in the sense that often non-European developments are simply not mentioned. But when they are mentioned it is often pure fanboyism. 82.135.75.49 01:34, 11 May 2007 (UTC)

Packing real numbers together

You can "pack" any finite number of integers or rational numbers into one (e.g. by alternating digits), but is this legal for real numbers? Fredrik | talk 02:47, 20 Feb 2005 (UTC)

Yes you can do it with real numbers too, at least when they are all in the interval [0, 1).

Start by writing each of those numbers in the form 0.a1a2a3…{\displaystyle 0.a_{1}a_{2}a_{3}\dots }, and for uniqueness, suppose that none of your numbers after a while contain only 9's.

Then, write the first digit of the first number, then the first digit of the second number, all the way to the first digit of the last number. Then work on the second digit of each number and so on.

Now, presumably you could do same thing without the assumption that the numbers are in [0, 1), but then you need to worry in addition about the exponent. Oleg Alexandrov 19:12, 20 Feb 2005 (UTC)

I've taken no number theory classes, so I may very well be wrong about this, but if i'm reading http://mathworld.wolfram.com/AlmostAll.html correctly then "almost all" can also mean "all of an uncountable set but countable number". —Miles←☎ 19:50, May 23, 2005 (UTC)

Come on guys - youre arguing the number of angels on a pinhead here -this is supposed to be an article for GENERAL consumption - not a finely hairsplit piece that is unreadable. the objection ""Almost all" is sometimes used synonymously with "all but finitely many"; see almost." is quite irrelevant, first because its one mans opinion, and secondly because it says SOMETIMES. It may SOMETIMES be synonymous with all but finitely many, but presumably that means that also sometimes it is NOT synonymous with that statement. "Sometimes may refer to;Sometimes - a pop song by Britney Spears Sometimes - an indie rock song by Ash Sometimes - a song by Hooverphonic" Almost all means to me "very many, but excluding at least some" Somehow I doubt that my definition is provably incorrect. Almost means "not quite"; the question for you is whether "All" implies a finite quantity. Is it possible to say "All numbers" If not, then youre right. If yes, then "almost all" in this context means an infinity, less a few. Thats quite a nice philosophical question though; can an number be infinite if some members of the class are excluded? Surely excluding a few would put finite limits on the set? i.e. is the set "All numbers except those between 0 and 1" finitely limited (at 0 and at 1)? Or is it infinite (because the upper and lower limits do not exist)? But I suggest this is limited to the talk page, and not included in any way in the general page.

This is a mathematics article; if you want an article on vague ideas people may have about real numbers you need to start a new one. In a math article, "almost all" means what it would mean to a mathematician; in this case, that the set of exceptions is of lower cardinality. Since the set of algebraic numbers is countable and the set of real numbers Powerset(countable), almost all reals are transcendental. That's what it means. It doesn't matter what it might mean when talking about Britney Spears, because it isn't talking about Britney Spears, it's talking math. Gene Ward Smith 06:58, 12 May 2006 (UTC)

I'd normally take it to me all except members of some set whose measure is zero. That doesn't mean lower cardinality. But of course in some contexts it means any of several other things. Michael Hardy (talk) 22:38, 23 August 2010 (UTC)

That’s a very snobbish response Gene. An encyclopaedia article on mathematics should be accessible to anyone regardless of background. You confuse formality with effective explication. An encyclopaedia article on mathematics that is only accessible to mathematicians is pointless as mathematicians can always read maths texts. Please keep this in mind in future.

Almost all links : Measure theory and Lebesgue. Almost all, all, but a negligible set; where a negligible set is a set with measure 0.v_atekor 12:44, 20 June 2006 (UTC)

I am a practicing mathematician (real analysis and probability theory). In my experience, "almost all" is generally used informally, to say that the set of exceptions is small in some sense which is usually understood from context (e.g. finite, countable, measure 0, nowhere dense, etc). However, "almost every" and "almost everywhere" have the precise meaning that the set of exceptions has measure 0, where the measure in question is to be understood from context, and in the case of real numbers, is Lebesgue measure unless otherwise specified. Since this is the sense intended in this article, I've edited accordingly. 128.84.234.217 (talk) 21:46, 23 August 2010 (UTC)

Question

how I edit the abstrct Which part of this is wrong:

A real number is the sum of an integer and a fractional part which has countably infinite digits (let's work in binary, so countably infinite bits).

The integral part can be one of ℵ0{\displaystyle \aleph _{0}} values, and the fractional part has countably infinite binary "choices" (i.e. there are 2ℵ0{\displaystyle 2^{\aleph _{0}}} fractional parts)

Thus, the cardinality of the reals is ℵ0⋅2ℵ0{\displaystyle \aleph _{0}\cdot 2^{\aleph _{0}}}, not 2ℵ0{\displaystyle 2^{\aleph _{0}}}

Equivalently, a real can be considered as a mantissa (in the programming sense) of infinitely countable bits and an integral exponent (in the same sense) --An amateur mathematician: Taejo 10:28, 7 August 2005 (UTC)

You've discovered that ℵ0⋅2ℵ0{\displaystyle \aleph _{0}\cdot 2^{\aleph _{0}}} = 2ℵ0{\displaystyle 2^{\aleph _{0}}}. Infinite cardinal arithmetic is funny that way. By the way, new stuff should go at the bottom of the page. Josh Cherry 12:35, 8 August 2005 (UTC)

I see that, but I was hoping for an explanation or proof. It seems to me that 2ℵ0{\displaystyle 2^{\aleph _{0}}} is the cardinality of a double-closed subset of reals: e.g. [0;1] --Taejo 21:42, 19 August 2005 (UTC)

It is, but that's the same as the cardinality of all the reals. Perhaps the following will help.

It's easy to make a one-to-one correspondence between a subset of [0, 1] and all the reals, which shows that the reals are no larger than [0, 1]. Take, for example, one cycle of the tangent function and dilate and translate it so it fits within [0, 1].

You point out that the cardinality of R is the same as the cardinality of N × [0, 1]. It should be intuitive that this is no larger than the cardinality of [0, 1] × [0, 1]. But the cardinality of [0, 1] × [0, 1] is 2ℵ0{\displaystyle 2^{\aleph _{0}}} (the total number of "bits" in the components of an ordered pair is countable).

Fundamental mistakes?

I refer to the sentence: 'Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.'

Well, the term real numbers has actually been used like a generic term just excluding imaginary and complex numbers. However, this subordination is at odds with fundamentals behind set theory. The second diagonal argument by Paul du Bois-Reymond / Georg Cantor provided evidence for the set of real numbers to be uncountable while Cantor also showed that the set of rational numbers is countable. In other words: According to CH, real and rational numbers are fundamentally different from each other. I doubt that there is a convincing explanation for this ancient discovery of 'alogos' so far. At least Ebbinghaus et al. wrote: '...it is possible to perform analysis without to know what makes the real numbers special (was reelle Zahlen wirklich sind)'.

Cantor claimed to use real numbers within his list. These numbers did not obviously differ from rational numbers except for two details. At first, the evidence demanded to change actually all infinitely many numerals. This is impossible to fulfill. Secondly, his evidence was given by contradiction, i.e. from the fictitious actual-infinite point of view.

Anyway. The pertaining real numbers are just the opposite of the quality to be real in that, they have to be fictitious solutions of a task that cannot be performed by a finite number of steps. Having reached that clarity we should remove misleading obstacles of understanding:

i) The continuum of fictitious real numbers cannot directly include rational and natural numbers. For instance, die embedded number one has to have an actually infinite number of numerals (zeros) behind the dot in order to be exact.

ii) Irrational numbers like pi must no longer be considered a single element but rather like the unrealizable task to have all representing numerals, e.g. all decimals. The number pi is as fictitious as the quadrature of circle.

iii) Be not mislead by stupid arguments like 'all numbers are fictitious'. One could likewise argue 'all numbers are imaginary'. Already Leibniz called infinitesimal numbers 'well-founded fictions'. The definition of real numbers by Meray 1869/ Cantor 1871 referred to fictitious limits of convergent sequences.

iv) Fictitious real numbers correspond to the amorphous genuine (non-Hausdorff) continuum where any finite amount of finite numbers is of no weight and correspondingly distinction between open and closed intervals is not justified. Also |sign(0)|=1.

v) It is true, real numbers are not apt for immediate numerical calculation. However, this restriction is already known. Admission of it does not cause any disadvantage. On the contrary, it will hopefully pave the way for some overdue minor corrections to mathematics.

Eckard Blumschein, Magdeburg, Germany.

Can you use less and better wording to make your point? There is nothing wrong with that sentence you refer to, it just says something like that some apples are red and some are not red. Oleg Alexandrov (talk) 07:56, 19 October 2005 (UTC)

The sentence misleadingly mingles two mutually excluding qualities: Rational numbers inclusive of natural and negative numbers are countable, while irrational numbers inclusive of transcendent and algebraic irrational numbers are uncountable. Since real numbers were shown to be uncountable they must not directly include rational numbers but merely embedded rational numbers. The latter differ from ordinary rational numbers in being just fictive numbers without complete numerical identity as are more obviously irrational numbers that correspond to unrealizable tasks like quadrature of circle. Embedded rational numbers also belong to a task that cannot be fulfilled: to represent an actually infinite amount of numerals. They are also outside the realm of finitism.

Moreover, it does not make sense to include or exclude any single real number from the continuum, not even the number 0.000...(actually infinite amount of numerals) is special. --unsigned post by Eckard Blumschein, Magdeburg, Germany.

You can use four tildas to sign your name, like this: ~~~~. Try it! And making an account will not hurt either, and take only 15 seconds of your time.

(EB) Thank you. I will try and obey it.

Now, who cares if the true rational numbers are merely embedded and not actually contained in the real numbers?

(EB) Let's clarify. A real number is p/q. An embedded real number is a*p/a*q with 'a' being 'actually' infinite. In other words, the embedded one is a 'real' number because it is just a fiction. Admittedly the wording is a bit silly due to lacking understanding of those who coined the words. If we ascribe properties to IR, then these properties depend on the fundamental peculiarity of all real numbers: All real numbers including the embedded rationals are just fictions. For that reason, they behave quite differently as compared with true rational numbers.

Using your terminology, there is a one to one correspondence between "true" rational numbers and their "fictive" counterparts on the real line.

(EB) I do not think so. Bijection is based on exact numerical identity. There is no chance to pinpoint a bijection between pi and any numerically addressable position at the real line. We are sure that pi is located at the real line but we cannot exactly localize a corresponding point. Real numbers correspond to fictitious points. Not even the bijection between pi and pi is a bijection between real numbers because the symbol pi just defines a task. The pertaining real number is the fictitious solution to this unrealizable task.

That correspondence preserves all the properties you could imagine of the rational numbers,

(EB) I agree with the mere word correspondence while objecting against the unproven and wrong guess 'one-to-one'. The very idea of actual infinity contradicts to the possibility of discrete numerical identity. Real numbers have a basic peculiarity. They are uncountable stuff constituting the continuum. In that they are fundamentally different from the discrete and countable rational numbers. Cantor was correct when he called infinity an abyss. As supposed with CH, there is no bridge between the two worlds discrete numbers and continuum. All effort aiming to use equivalence, isomorphism etc. is doomed to fail.

and so, who cares if the real line contains "fictious" or "true" rational numbers? Oleg Alexandrov (talk) 14:50, 19 October 2005 (UTC)

(EB) Since even the tiniest interval of the real numbers is commonly imagined to contain an actually infinite amount of numbers, one has to ask for logical consequences. I conclude: Single real numbers, if they did exist at all, would absolutely not matter. The same ist still true for the hugest imaginable finite 'subset' of reals. Perhaps you will already comprehend that embedding of true rationals does not make any difference to the entity of reals. Nobody may care of them. They are lost. Consequently, the real line of continuum does not exhibit anything else than fictitious numbers while the other way round, the real line of discrete numbers merely contains discrete numbers.

By the way, I agree with the statement: "The reals are uncountable". However, I do not share Cantor's interpretation claiming strictly more real numbers than natural numbers." I argue: 'Countable' and 'finite' are qualities, not quantities. Being fictitious and being uncountable are just two aspects of the same peculiarity that makes the real numbers essentially different from the rational ones. We have to abandon the notion of isolated points in case of the continuum IR. So the basis for counting is missing there. Do not bother about that. It does not harm analysis.

(EB) I disagree. The decisive peculiarity is to consider the fictitious limit of this sequence. As long as you do not claim to completely include actually all of the infinitly many terms, the Cauchy sequence is still a rational number. Being impossible makes the difference.

One can embed the rationals in the reals by the map q-> (q, q, q, ...).

(EB) Really? This would demand numerically addressable points at IR.

It seems that by real numbers you mean something else.

(EB) I refer to what Cantor demonstrated to be uncountable. I also refer to the continuum.

Let us stick to the axiomatic approach please,

(EB) Sorry, this would not fix the problem but perpetuate the errors. As found out by Brouwer: The law of excluded middle is basic for discrete mathematics, but it is not valid for the infinite.

otherwise we are getting into metaphysics which is outside mathematics.

(EB) Metamathematics rules mathematics. Selfconsistency of mathematics demands that uncountable numbers/sets must not be mingled with countable numbers/sets. Well, we need a fundamental correction to the abundance of axioms.

The real numbers are defined as the set of all infinite Cauchy sequences of rational numbers, where two sequences are considered the same if their difference has zero as limit. This is the definition. There is nothing you can do about it. Oleg Alexandrov (talk) 11:32, 21 October 2005 (UTC)

I regard this definition absolutely correct. It dates back to Meray (1869) and Cantor (1871). Other definitions (Dedekind cuts, nested intervals, equivalence classes) were claimed to be equivalent. I would like to object that there is just a single decisive criterion: In order to really define real numbers, not rational ones, the word infinite has to be taken seriously. This can be stressed by the attribute 'actual'. Of course, there is no actually infinite size. Infinity denotes a quality. One cannot make it larger or smaller. Cantor's quantitative notion of infinity was a big and tragic mistake. Nonetheless, Cantor's second diagonal proof assumes the real numbers in accordance to the definition with emphasis on actually infinite sequences.

On the other hand, Dedekind's proud cut fails to actually provide any real number because Dedekind already started with the incorrect supposition of TND.

So, given that we agreed on the definition of real numbers, why isn't the map r→(r, r, r, ...) from the rationals to the reals a good embedding? Oleg Alexandrov (talk) 00:09, 22 October 2005 (UTC)

A zealot or fundamentalist. But not able to express his point of view. Or to cite anyone able to express his point of view. I had almost exactly the same discussion, using the same arguments, with (EB) on the german page.--LutzL 15:46, 17 February 2006 (UTC)

EB appears to be an advocate of what is actually quite an interesting philosophical viewpoint (although one that is uncommon among mathematicians): the rejection of the infinite. At one point he complains that "They are also outside the realm of finitism". Finitism (of which Kronecker was the most prominent advocate) does not believe in objects that take an infinite number of steps to construct from the natural numbers. However, Reuben Goodstein was able to derive results of real analysis within this viewpoint. It is well-known that constructivism (of which finitism is a more extreme version) is just a good foundation for mathematics as the looser Zermelo-Fraenkel set theory which is more widely assumed (perhaps better - see List of statements undecidable in ZFC). Apparently the ultrafinitism of Alexander Esenin-Volpin is an even more extreme version of this philosophy which denies the existence of the natural numbers as an object, as they take an infinite number of steps to construct. To me, this seems a more tenable philosophical position than finitism: why permit a countable number of steps to construct the natural numbers, but then reject using a countable number of steps to construct anything else? Unfortunately, ultrafinitism is reported to be inadequate to do much mathematics of interest.

The fact is, in virtually all mathematics we discuss facts about infinite sets about which we have less intuitive reason to believe make sense than perhaps we have about combinatorial, finite objects. When one accepts the result that there are an infinite number of statements that one can make about the natural numbers that may be taken as either true or false as axioms, one realises that the whole of mathematics is not quite as solid as it may have seemed before Gödel. However, the translation of all statements about the real numbers into statements about sets of pairs of natural numbers (i.e. rational numbers) through the Dedekind cut construction shows that there is no reason at all to feel less comfortable believing in the consistency of real analysis than that of number theory. Elroch 19:50, 17 February 2006 (UTC)

Axiomatic approach

In the present form, this subsection does not make sense to me since its a mix of definition and theorems. Probably, the author meant something in the lines of

Suppose given a structure (R,+,⋅,<){\displaystyle (\mathbb {R} ,+,\cdot ,<)} with the following properties: […{\displaystyle \dots }] Then (R,+,⋅,<){\displaystyle (\mathbb {R} ,+,\cdot ,<)} is a model for the real numbers.

I did not see any definitions and theorems in that sections, they all look like axioms to me. They are a bit informally stated, but they are axioms nevertheless. Oleg Alexandrov (talk) 00:35, 21 October 2005 (UTC)

Subsets of the set of real numbers

Maybe we should explain the uses of R+, R-, R0+, R0- as special subsets of the real number set R

I would agree with that, in a section say before "Generalizations and extensions". Oleg Alexandrov (talk) 22:33, 7 December 2005 (UTC)

The problem with this is that the notation is not completely standard. If R+ appeared in a math paper, for example, it wouldn't necessarily be clear if it's supposed to contain 0 or not. The interval notation (0,∞) is more widely used and requires no explanation. Brian Tvedt 12:03, 12 December 2005 (UTC)

Just because notation is not always well-defined is no reason to not write about it. One could still define it and explain the various interpretations of that notation. However, I would agree that I did not myself see this notation often, and I am not sure how valuable it would be in the article. Oleg Alexandrov (talk) 16:07, 12 December 2005 (UTC)

0.999* = 1.0

Recent edit war does highlight an important point. The question of whether 0.999… = 1.0 is one of those common questions which comes up when talking about the reals. Currently the article only contains the statement (for example, the representation of 1 is 0.999…). without any further treatment or links to appropriate places where its discussed. As one of the those recurent questions I feel it should get a better explination. Who first proved it, or is it axomatic? (I seem to recal there is some set where 0.999… != 1.0). --Salix alba (talk) 20:47, 16 February 2006 (UTC)

The article Proof that 0.999... equals 1 seems well written and shows that the edit I reverted was merely a bad link :-). On a related point, it seems to me that Real numbers should say "Every real number has a unique representation as an infinite decimal not ending with an infinite sequence of 9s", like the article Decimal. This avoids the ugly exclusion of 0, and I think most people would be inclined to choose representations like 5.000... as canonical, rather than 4.999... Elroch 22:49, 16 February 2006 (UTC)

I honestly can't see whats wrong with a simple link to the article in a see also section. Note this is not a theorem about base 10 numbers per say, it is a theorem about the representaion of the reals in any base. Seeing as real numbers are technically defined as an infinite sequance of digits in some base, this seems a perfectly reasonable statement. I'm sure cantor would have something to say on the subject. --Salix alba (talk) 14:20, 21 February 2006 (UTC)

Actually, that's not the way the reals are usually defined in rigorous treatments. You could, if you wanted to, and everything would come out all right in the end, but you'd have a lot of technical annoyances along the way (try, for example, defining multiplication on reals thus defined, and then showing it's associative). The usual way to define them is via Cauchy sequences of rationals, though the older method of Dedekind cuts still has advantages in some contexts. --Trovatore 19:00, 21 February 2006 (UTC)

This situation will occur in any number base, not just base 10. If B, a positve integer greater than 1, is used for our base, then .[B-1][B-1][B-1]... (.[B-1] repeating)is the same as 1. In base 2 we have .111... = 1/2 + 1/4 + 1/8 +...= 1. What is the difficulty with accepting .9 repeating is 1?

definition

Nowhere could I find a decent definition of what real numbers were in this. I might have missed it. At any rate, I added

"Formally, the set of real numbers is the closure of the rational numbers"

err I should add that I realize it says "they can be constructed..." but I think it should start out by saying something like this since its the definition. --anon

Its not just any closure, its the closure with regards to the eucliden metric. There are other closures of the rationals, called the p-adic numbers. linas 15:29, 22 March 2006 (UTC)

As a scientist, but not a mathematician, I roughly know what a real number is - I think, because I havent been exposed to these distinctions for a while - and I needed a definition for some purpose, and came here. I now remember why I hated how math was presented in school and college. (I took it to 3rd year university). Although this page is full of esoteric details about the things you can do WITH real numbers, - which is fine if you know what a real number IS already - THERE IS NO INTRODUCTORY, GRASPABLE DEFINITION OF WHAT A REAL NUMBER ACTUALLY IS ON THE PAGE.

WHY are mathematicians so obsessed with making maths texts impossible to understand unless you already know the field? It seems to be a commonality.

The one statement that reassured me that my definition MIGHT be correct, was the statement that real numbers were retroactively defined in response to imaginary numbers! I dont want to mess up the page, as I'm no mathematician, so I wont change this myself. But can someone PLEASE introduce the following statement, with whatever changes are needed to make it accurate? The historical section is fine, if you dont want it at the top.

"The term "real number" is a retronym coined in response to "imaginary number". For thousands of years, all numbers used were real numbers. However while the square root of 1 is 1, and a real number, the square root of -1 does not compute. However this number - whatever it is - is useful, and has been called <i>. All numbers which are multiples of <i> are Imaginary numbers. (link to page on imaginary numbers). In recognition that there existed numbers which were not divisions of the regular number line from -OO to +OO, the regular numbers were renamed as REAL numbers, to be distinct from Imaginary numbers. All numbers that are not imaginary, are real numbers."

I mean come on guys - this is what its all about, right?

given that the term real number distinguishes only from real and imaginary, its appaling that this distinction is not given clearly on this page. Its like asking someone what non-existence is, and being told the many theories of life after death, properties of a vacuum, etc, instead of saying that its everything that doesnt exist.

Intutitive defintion

I don't like it, since it reeks of circularity. The only way we can related line to real number is by constructing the real numbers and relating that to lines. So far as synthetic geometry axioms go, the axioms could just as well have a model using other numbers, such as recursive numbers or hyperreal numbers. Moreover, saying it is a line misses the field structure! Discussion? Unless this gets some sort of defense I'm changing it. Gene Ward Smith 01:48, 10 May 2006 (UTC)

Point taken, but the defence is that the introduction for a topic like this needs to be aimed at a very general audience. You can't use words like "construct", "axiom", "field", etc. It's okay if it's a bit circular, because most people aren't even aware that their intuitive concept of "number line" depends on a rigorous definition of "real number". The more rigorous stuff is later on in the article, where it should be. Dmharvey 02:28, 10 May 2006 (UTC)

It's the word "defined" that bothers me, I guess. Maybe another word would be good. Gene Ward Smith 04:08, 10 May 2006 (UTC)

"It's the word "defined" that bothers me, I guess. Maybe another word would be good." How about "intuitively understood"

Again guys, speak to the person who needs this page, not the experts. I like the point about the GENERAL AUDIENCE, but again, youre arguing fine points, instead of the main definition. THere is a problem in talking about the number line, in that presumably the numbers from -oo i to +oo i are also part of a number line. An imaginary number lne. Field structures are meaningless to the reader, unless youre talking reproducing whitehead and russell or something. REALS are NOT IMAGINARIES. and THATS hat MAKES REALS, REAL. Please keep the properties out of the definition, unless you can find a property which distinguishes clearly between Real and Imaginary. The extra stuff can be added in later on; ONCE the definition has been correctly made.

Sorry Shantrika, I have to disagree. Saying that "reals are those things which are not imaginaries" is grossly misleading. There are lots of things that real numbers aren't. For example, they're not p-adic numbers. Your definition is a bit like saying "a laptop is any machine that runs on batteries" -- it assumes the reader thinks that all machines are computers. And I certainly don't think all our readers think that all "numbers" are complex numbers. Dmharvey 11:13, 12 May 2006 (UTC)

Gaps and stuff

I reverted the rewrite of the intro by Gene. In it the word "gaps" shows up three times, and I really think that continual mantra on how the rationals are riddled with holes barely wating to be filled is not the right tone to start this article with.

It makes much more sense to start the article the way it is now, post-revert, with the numbers being a one-to-one correspondence with a line. That, if you wish, says the same thing, the reals are a continuum, the way a line is, but in a much more approachable and elegant way.

It makes no sense to leave the article as it is, because as it is is wrong. The point is to explain what real numbers are, and this fails to do that.

Some of the stuff I reverted is perhaps salvageable. We can get to it after the introduction business is settled. Oleg Alexandrov (talk) 04:51, 13 May 2006 (UTC)

I think you should talk more and revert less. I don't agree with the gaps. Let us see what others say. Oleg Alexandrov (talk) 20:47, 13 May 2006 (UTC)

I quite like the intent of Gene's edits, maybe not with the wording though. I think it's a bit heavy on the "ordering" language. Dmharvey 22:13, 13 May 2006 (UTC)

How do you propose to discuss the fact that the real numbers are a continuum without circularity, unless you bring in order? The usual way to define or toconstuct an example of a continuum is by using the real numbers, for example in constructing the long line. Gene Ward Smith 22:54, 13 May 2006 (UTC)

I don't see what's "mathematically incorrect" about "Oleg's version" (this diff). And if it were incorrect, how could Gene make it correct by adding more stuff? Anyway it's not really true that there's "no room" to put more points into the real line; for example, you can embed it into "the" hyperreals (scare quotes on "the" because they're not unique). Gene, I think you need to calm down a little. Saying the reals can be "thought of" as being the elements on a line doesn't express anything terribly precise, perhaps, but it's not wrong, and it's probably helpful to the naive reader. --Trovatore 22:32, 13 May 2006 (UTC)

Oleg's version is mathematically incorrect because any totally ordered field, and certainly any totally ordered field embedded in the reals, has a "line". Hence, it isn't an intuitive definition, and I think that makes it unacceptable on the grounds that it's simply wrong. As for the hyperreals, you missed how I pointed out that you can add more to the rationals and still keep the field structure. With hyperreals, you have a "gap" between the finite hyperreals and the positive and infinite hyperreals, but you can't fill in the gap and complete the field. Hence, what I said was correct. And no, I don't think it is helpful to the reader to tell him or her that the reals are uniquely able to be thought of as a line, because that's false, and false is bad. Gene Ward Smith 22:40, 13 May 2006 (UTC)

Where did the word "uniquely" ever appear? The text said that the reals could be thought of as a line. That's true, not false; "uniquely" is your own interpolation, and no one's claiming it's a "definition", intuitive or otherwise. (Note that there's no requirement that a definition appear in the lead section; the lead section is for summarizing, not defining.)

I don't agree that any totally ordered field embedded in the reals are a "line"; in particular, the rationals are not intuitively a "line". As for completing the hyperreals, you'd have to say what sense of "complete" you're interested in; I assume Cauchy complete, but what I found in the article talked about the Archimedean property, which your lead-section text said nothing about. You can find "room" to put more points between the reals. You lose the Archimedean property, and probably lots of other properties, but you can do it. So your text is "false". --Trovatore 05:47, 14 May 2006 (UTC)

Of course the rationals are a line. They not only get called an (affine) line, in contexts such as AQ1, but as totally ordered, they look just as much like a line as the reals do, because both look the same. Moreover, some people profess to be perfectly happy stopping at the computable reals and using them in place of the reals, and that implies presuming they are a line. If you used the computable reals to construct a model of Euclidean geometry, it seems to me the lines you get are perfectly honest lines. Could Euclid have told the difference? Gene Ward Smith 07:14, 15 May 2006 (UTC)

No, I disagree. Intuitively they're not a line, but a "fog" or something. If you tell people the real numbers correspond to distances along a line, they'll know what you mean, without formalities (though they probably won't be able to formalize it themselves without help). It connects to a physical intuition, which is what's needed here, rather than all the stuff about gaps and ordered fields. Telling people there are no gaps doesn't add much; of course there aren't gaps in a line. If there were gaps, it wouldn't be a line. All that language does is increase the opportunities for confusion. --Trovatore 07:30, 15 May 2006 (UTC)

I have no idea what a "fog" is. Are the computable numbers a "fog"? As I pointed out, that's what some people are using by way of a line. Are hyperreal numbers a "fog"? They allow for models of Euclidean geometry also. As for "physical intuition", this seems like a Kantian position--we have a built-in notion of space, what Kant called a "form of intuition". That's not good enough for mathematics, because if it doesn't tell us what the properties of a line are well enough to contruct the reals, then the intuition is misleading us. As for "of course", in mathematics there is no "of course". There are no gaps if we insist there not be any, having first defined them, but geometers aren't all saying what you seem to claim they ought to. The affine rational line is a perfectly fine line for many purposes, and for some purposes where it won't work, other fields besides the real numbers often will work. Algebraic geometry over the algebraic closure of the rationals works pretty well, for instance. Gene Ward Smith 08:10, 15 May 2006 (UTC)

I think we do have a built-in notion of space, and when elaborated correctly, it results in the reals. It does tell us what the properties of the reals are, well enough to construct them; it just takes a bit of work.

That's what we should be trying to connect with in naive readers. Yes, of course the computable numbers are a fog; there are only countably many of them, so they have to be. There's nothing wrong with these other structures, but they aren't "lines". --Trovatore 15:01, 15 May 2006 (UTC)

I put this on the WikiProject Math talk page:

Incidentlly, so far as calming down goes, Oleg reverted my page without first reading it, and then insulted me for not writing stuff he didn't bother reading. I think he is a little out of control here, and needs to calm down. Gene Ward Smith 22:42, 13 May 2006 (UTC)

Ok, since it seems people here are waiting on some outside opinions...Gene's first version with stuff on gaps was inappropriate for the intro. It was too wordy in any case. I like the "original" version better. As for the current version, which adds the statement about "property that there is no room in the line to fill in more numbers", that seems ok to me. --C S(Talk) 03:34, 14 May 2006 (UTC)

Except of course as Trovatore points out, there is room to fill in more numbers. -lethetalk+ 05:49, 14 May 2006 (UTC)

"No room" was not my preferred language; I introduced it to appease Oleg, who seemed to have an allergic reaction to "gaps". However, the article on continuum (mathematics) says "gaps", and I'm now linking to that. "No room" was intended to convey the idea that there weren't any gaps in which you could stick more numbers, ie, that the reals form a continuum. Which is true, but if the language is worse than it originally was, go back.

The stuff on "gaps", explaining what a "gap" is, is now found in the "basic properties section, which people keep destroying with these reverts.

Finally, I note that Lethe has gone ahead and reverted without discussing, even though there had been a gentleman's agreement not to do that. The reversion was based on blaming me for language I put in to appease Oleg, and which not everyone found objectionable, and demolished a lot of material without discussion. Is this not turning into reversion for the sake of reversion? It seems to me there is an attitude here of "I'm an administrator, so I get to decide". But that's not policy. People's adminstratorship does not add any lustre to their knowledge of mathematics or their writing skills. Please let's have a proposal before arrogantly reverting. Gene Ward Smith 07:03, 15 May 2006 (UTC)

I disagree wholeheartedly that an attitude of "I'm an administrator, so I get to decide" is present. Administrators are sometimes stubborn, and non-adminstrators are sometimes stubborn. -- Jitse Niesen (talk) 09:43, 15 May 2006 (UTC)

No, I noticed that right away, but I wouldn't say that intuitively the result is a "line". Of course, it's all very vague wording, which is why I only said I was "ok" with it. I would stick with the "original" wording in any case. Despite Gene's concern over the possible misinterpretation of the intuitive stretch being made, in my experience (which admittedly is mostly university teaching experience) I've never found anybody to actually be confused over considering the reals as a number line. --C S(Talk) 06:24, 14 May 2006 (UTC)

I'm concerned that it seemed to have been presented as an intuitive definition, which I think is a bad plan. Gene Ward Smith 07:03, 15 May 2006 (UTC)

Comparing the version before the childish edit war with the current version, I think that the former version does not make the difference between the rationals and the reals sufficiently clear. Gaps seem to be a good method to explain this; I think that is more accessible than the property that each bounded set has a least upper bound. So, at least some aspect of this explanation should remain, in my opinion.

However, much of the section "Basic properties" currently duplicates stuff further on. My suggestion would be to keep the "ordered field" stuff out of that section. Another approach to explaining the difference between the rationals and the reals is to say that the reals are infinite decimal expansions. That seems intuitively pretty clear to me; what do people think about that?

Finally, I do not like the mentioning of gaps in the first sentence ("In mathematics, the real numbers may be thought of as numbers that correspond to the points on an infinite line — the number line — which has the property that there are no "gaps" in the line; the real number line forms a continuum."). This is rather confusing, I think, because the gaps are actually quite strange gaps. I fear that somebody who does not already know about these gap won't understand this sentence at all. -- Jitse Niesen (talk) 09:43, 15 May 2006 (UTC)

What do you think of saying in the first sentence that real numbers are numbers which can be expressed via infinite decimal expansions? I think you are right that infinite decimal expansions are likely to be thought of as intuitive, and after all, the real numbers can be rigorously constructed in that way. Then, having said that, use the magic word "continuum" and explain what that means intuitively as a lack of "gaps", which is further elucidated in the basic properties section? Gene Ward Smith 09:54, 15 May 2006 (UTC)

Real number

User:Oleg Alexandrov seems to me to be engaging in abusive reverts on this page, to a previous verison which is arguably incorrect mathematically and which removes a lot of new material, material for which he has given no argument for removal. He also says, falsely, that my attempt to satisfy his previous criticims amounted to "writing a one-liner" which seems to prove he hasn't even seriously looked at the version he is reverting from. I think we need other people to weigh in at this point. I am very much opposed to simply allowing it to say the real numbers have a number line and calling that a definition. My proposal to say they have a number line, with no "room" to fit additional numbers in, is an attempt to make the one-line introduction correspond to an actual rigorous definition, which will not be the case if we allow Oleg's revert. Gene Ward Smith 22:32, 13 May 2006 (UTC)

That was not "my revert". That was the version of the page which has been here for a long while, and which presumably reflected the community consensus. You had been and continue making huge changes to this page without sufficient discussion and using improperedit summaries, instead of waiting for consensus to develop on the talk page. Oleg Alexandrov (talk) 23:06, 13 May 2006 (UTC)

Unlike you, I did open up a discussion before making changes. It was clear as a result of that that there was no such consensus as you imagine. You, on the other hand, unilaterally reverted, and then re-reverted, without first discussing the matter. As for my edit summaries, summarizing an edit in one line is not easy, but I've been trying. If you want now to explain why you reverted and re-reverted, I suggest you do so, but saying something is "ugly" in an edit summary is not an explanation, and hardly puts you in a position to criticize other people's edit summaries. Gene Ward Smith 00:19, 14 May 2006 (UTC)

The consensus I see above was to keep the introduction simple and intuitive. I reverted your changes which made the introduction very complicated and explained my revert on this talk page, right above. You chose to reply with the cryptic "We can change the wording" then reverted my revert and plowed ahead with even more changes and additions.

However, taking into account that you are relatively new and I have been here for a long while, I admit that I should not have reverted to the original to start with, at least not the second time.

There seems to be little support from the other editors for the article the way you rewrote it, although you may have a point about making things more rigurous. I suggest we put mutual accusations behind us, and after a period of cooling down try to hammer out a version acceptable to everybody. Oleg Alexandrov (talk) 00:58, 14 May 2006 (UTC)

Can I suggest you turn your attention to the rational number page, its in a pretty poor state and could do with a bit of attention. Improvements there are likely to be better received than in the fairly stable article here. --Salix alba (talk) 23:17, 13 May 2006 (UTC)

Perhaps you could stop making allegations about Oleg. He's not being abusive, he's just being conservative about additions to important articles. He does this for a lot of articles, ensuring that big edits are actually improvements, instead of just adding words. Articles that don't have people watching over them get long and filled with rants and rambles. Assume a little good faith. Oleg has been here a while, and I trust his editorial decisions implicitly. If you want to make changes that others disagree with, why don't you propose them on the talk page, and defend them, instead of starting an edit war. -lethetalk+ 05:46, 14 May 2006 (UTC)

Thank you Lethe. Well, the fact of the matter is that I should have shown more consideration. I do hope there will be some agreement about how this page looks like. Oleg Alexandrov (talk) 16:14, 14 May 2006 (UTC)

I've contributed a lot to Wikipedia, and to the mathematics articles in particular. I think it should be presumed that I am operating in good faith and thought should be given before reverting. Probably anyone with PhD in a relevant field deserves at least this much for any Wikipedia article, and if that person has been around a while and contribited a lot, so much the more. Oleg has these sort of credentials, but so have I, and I suspect I've done more in the way of adding actual content. My focus has not been administrative, but that does not make me a second-class editor. Gene Ward Smith 03:02, 15 May 2006 (UTC)

Alright, I apologize if I came across as rude. Certainly you deserve respect as a valued mathematical contributor. You see how making accusations of bad faith can cause people to get defensive. It's simply counterproductive. Discussion is always welcome. In the meantime, I stand by my reversion. -lethetalk+ 10:48, 15 May 2006 (UTC)

While Lethe's comment was a bit too defensive of me perhaps, the edit summary has objective information for why the revert was done. I suggest we do not do any more reverts, as it just makes people miserable. I suggest to let things cool off for a while, then start a genuine discussion on this talk page on how to improve this article (without huge changes to it in the meantime). Oleg Alexandrov (talk) 04:04, 15 May 2006 (UTC)

I have reverted to the version with wider support (before reading Oleg's suggestion above, which I consider untenable for opinionated edits). However, there are things that need improving. In my opinion, the introduction does need expansion, especially as the idea of it being characterised by being a "line" is so vague and, as pointed out in discussions above, inaccurate to a purist, without using something close to a circular definition. I think the argument for giving some intuitive idea of connectedness or completeness in the introduction is a good one, and this is the part of the last edit I felt least inclined to revert. Surely some agreement on what more is needed, and how it should be worded is possible?

Also it might be worth having more on relationship between the least upper bound property, the Archimedean property and the lack of infinitessimals, as these are key to understanding what characterises the real number field. Elroch 09:59, 15 May 2006 (UTC)

Obfuscating edits

Gene, you have on more than one occasion deleted the following clear and precise sentence, and the following elaboration, and replaced it by a rambling and far less clear paragraph.

Please do not repeat similar changes unless you can find substantial support here. Elroch 11:01, 15 May 2006 (UTC)

You are one of the characters who has been reverting while ignoring the ongoing discussion, without first attempting to find support, so I'd say you are not in a good position to complain. If you think an edit is rambling, I suggest you either re-edit or make that point on the talk page. As for that sentence, obviously there's nothing wrong with it but I think I kept in in some form. Gene Ward Smith 17:04, 15 May 2006 (UTC)

If it comes to a vote, or an RfC, I side with Elroch over Gene Ward Smith in terms of keeping in mind the "ongoing discussion". So far, no one agrees with you (GWS) that your definition should be in the introduction. — Arthur Rubin | (talk) 17:10, 15 May 2006 (UTC)

Introduction

The introduction of general articles is perhaps the most difficult part, as it needs to be accessible to someone with inadequate background to understand the rest of the article, but which should not be misleading. This seems a difficult task in this article: at present the introduction is intuitively clear, but does not make any attempt to identify what properties characterise the real numbers.

The introduction needs the inclusion of an intuitive mention of a property that distinguishes the real numbers among ordered fields. The two obvious alternatives are the least upper bound property and connectedness. How may one of these be described to give an accurate picture to a non-mathematician? (the word "gaps" alone would suggest a hole of finite size to most people)Elroch 12:08, 15 May 2006 (UTC)

The intro as you seem to prefer it is in fact not intutively clear, because it suggests being on a line characterizes the real numbers, and it doesn't. I tried out a first sentence which said the real numbers were the numbers with a decimal expansion, which is accessible to the mathematically uniinitiated, but that got immediately deleted also. What was wrong with it? It's mathematically correct, does in fact characterize the real number numbers, and can be understood by the nonmathematician, so of course it's immediately deleted and replaced with something which suggests something incorrect.

How do you explain the reals are connected? You can't use path connectedness, which would be circular. You can't very well use general point-set topology. I don't see how to make connectedness work, which is too bad. LUB is equivalent to Dedekind complete and to the "no gaps" property, and it seems to me of the three "no gaps" is the most intutitive. Gene Ward Smith 17:38, 15 May 2006 (UTC)

The description of the real numbers as infinite decimals is mathematically ugly, but the most widely accessible description. Hence if I saw this in the introduction I would cringe and leave it, as I did when I saw it later in the article.

The definition of connectedness comes from the natural order topology. It is worth pointing out here that your text about gaps was misleading because, as was pointed out, the rationals have no gaps in the mathematical sense, so this condition does not even distinguish the reals from the rationals. Hence your decision to repeat this edit was surprising. As I pointed out, connectedness is a stronger condition than having no gaps, distinguishes the reals among ordered fields, but is difficult to include in an intuitive introduction. Most people probably understand a line as being continuous in some sense (drawn without taking a pencil off the paper), despite the inevitable lack of formality in this, so that is a useful concept to include. Elroch 09:27, 16 May 2006 (UTC)

I don't think we need to get into that level of detail at first presentation of the concept. The thing to do is find language that connects with the physical intuition, the "built-in notion of space" as you call it. It really does exist, and it's the thing that's going to make the most sense to the naive reader. (Well, yes, some of them might connect with decimal expansions too, but that has the unfortunate side effects of suggesting radix dependence and of bringing up the point-nine-repeating thing; see Talk:Proof that 0.999... equals 1/Arguments for the worst-case scenario of that.) --Trovatore 17:46, 15 May 2006 (UTC)

I agree with Trov. Also, everybody knows what a line is. Gene's verion had the real numbers being a "line without gaps", which would be confusing, as lines don't have gaps to start with. Oleg Alexandrov (talk) 17:48, 15 May 2006 (UTC)

Since people are saying the rational numbers aren't a line, I don't see how you can claim there is any kind of clarity on the definition of "line". I didn't get answers to my questions--are the computable numbers a line? After all, there aren't any computable gaps. You can't computably determine two sets of computable numbers which make a Dedekind cut for a non-computable number. You can't computably find a set of computable numbers with an upper bound which does not have a least upper bound. Where's your intution about "lines" on this one? What about surreal numbers? If you find a gap, you just stick in another number; that's how they are defined. You can't continue to do that when they form a proper class, but that doesn't make the "Field" of surreals much like the real numbers. Is this a line? The trouble is still one of circularity--if by "line" you mean homeomorphic to the real numbers, you are going around in a circle. Gene Ward Smith 18:02, 15 May 2006 (UTC)

I did answer you—no, the computable numbers are not a line, they're a fog of points. Of course it would be circular as a formal definition to say that a line is something homeomorphic to the reals, but a formal definition is not what I want here; I want to connect with the underlying shared intuition. People already have the intuition; we don't need to create it for them. All we need to do is point them to it, so they can say "oh right, that".

Then it takes quite a bit of work to see that the underlying Platonic object that they intuitively understand is homeomorphic to the reals, and not to one of these other structures. But that's the right answer in the end, which is all that needs to be conveyed at the first presentation of the concept. --Trovatore 18:13, 15 May 2006 (UTC)

Personally, although appealling, I am not convinced it makes sense to talk about the underlying Platonic object as if it was a well-defined concept. Our belief in tbe real numbers comes from two things - the first is measurements in the real world, and the second is formal mathematics. I am not convinced that the set of sets of the rationals that are bounded above is an unambiguous concept (don't ZF and constructible set theory differ here?), although extrapolation of intuition from finite sets to infinite sets would suggest it is (I believe incorrectly). Elroch 09:44, 16 May 2006 (UTC)

Also, what on Earth is a "fog"? Incidentally, it is my understanding that the computable reals are computably connected, in the sense that any computable Dedekind cut defines a computable number, so we can't break them into two open sets computably. Elroch 09:52, 16 May 2006 (UTC)

Asides from your discussion on "gaps" I'd like to point out that grade schools (at least here in the U.S.) teach real numbers as the set that is the union of rationals and irrationals and that is likely where many readers are going to look at this from.
Also, why is the article organized as such: Basic Properties, Usage, History, Definition, Properties, Generalizations, etc.? I think may be we should work on restructuring the article and making it more clear before we add new stuff? (If we do add "gaps" we should merge it with Properties may be rather than sticking it at the top.) -- 127.*.*.1 00:44, 16 May 2006 (UTC)

I'd like to chime in again here. I think it's clear from the discussion so far that we all have quite different ideas about (1) what is the best way to think about the real numbers, and (2) how many, and which of these ways, should appear in the introduction to this article.

With this in mind, I think the introduction needs to have the following elements. It needs to have

Something along the lines of "the real numbers include both rational numbers and irrational numbers" (with examples), along the lines of what 127.*.*.1 suggests. My rationale is that average Joe will, after reading this sentence, actually have a pretty good idea of what this article is about. If we only talk about "gaps", or Dedekind cuts, or Cauchy sequences, or even "points on a number line", I still think average Joe might still not know what the article is about at all. Mentioning numbers like 7 and -119/47 and Pi and the square root of 2 is the quickest way to bring him/her onto the intended wavelength.

For the same reason, it needs to mention infinite decimals; this has the advantage of providing both an "accessible" explanation of real numbers, and can be massaged with some work into a acceptable formal definition.

It needs to mention that historically, giving a definition of "real number" that is suitably rigorous for the purposes of pure mathematics was not an easy task, and was one of the most important developments of the 19th century.

It needs to give some hint of the various formal definitions that are available.

In the end this makes for a long-ish introduction. I don't think that's a problem; it's a very important, central topic, and I think there really are a lot of things that need to be said upfront.

So here's my attempt:

In mathematics, the real numbers may be thought of in several different ways. From a "naive" point of view,

A real number can be given by an infinite decimal representation. For example 2.4871773339...., where the digits continue in some way, is a real number.

The real numbers correspond to points on an infinitely long number line.

However, these descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — was one of the most important developments of 19th century mathematics. Some of the more popular definitions may be summarised as follows (more detail given below):

It is possible, though cumbersome, to give a rigorous definition along the lines of "infinite decimal representation" as described above.

The real numbers may be obtained by starting with the set of rational numbers and "filling in the gaps" using the notion of Cauchy sequences. (The "gaps" here are the irrational numbers, but to avoid circularity of definition, one cannot define "irrational number" to be "a real number that is not rational", since the goal is to define "real number"!)

Again, starting with the rational numbers, certain kinds of sets of rational numbers ("Dedekind cuts") are declared to be real numbers.

The real numbers may be defined axiomatically as the unique complete ordered field. Although this approach does not provide a construction of the real numbers, it gives a list of properties that they must satisfy, which specifies them unambiguously.

Finally, I just want to ask, does anyone have a reference for the claim that "real number" is a retronym as described in the current text? Even better, it would be very interesting to know where the term first appeared. Dmharvey 12:06, 16 May 2006 (UTC)

This looks good, but I hope that there will be no bullets in the introduction. I mean I like the ideas you mention, but I hope they can be rephrased and put together into a few short paragraphs with the most important highlights. Oleg Alexandrov (talk) 15:00, 16 May 2006 (UTC)

This seems a good attempt to make the introduction accessible. One correction - the real numbers are the unique complete ordered field with the Archimedean property (or, rather neatly, the unique connected, ordered field). Also, I am familiar with Keith Devlin's use of the word gap, meaning two points with no point between them. With this meaning, a total order is connected iff it is order complete and has no gaps. However, I think Devlin's meaning is a "local" one, so unless someone else knows otherwise, I don't have any objection to a different use of the word here. Elroch 15:55, 16 May 2006 (UTC)

Oleg, the bullet points don't bother me too much, but let me see what I can do. Elroch, thanks for the correction. How about this:

However, these descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — was one of the most important developments of 19th century mathematics. Some of the more popular definitions used today may be summarised as follows (more detail given below).

First, it is possible, though cumbersome, to give a rigorous definition along the lines of "infinite decimal representation" as described above. Another is to start with the set of rational numbers, and to "fill in the gaps" using the notion of Cauchy sequences. (The "gaps" here are the irrational numbers, but to avoid circularity of definition, one cannot define "irrational number" to be "a real number that is not rational", since the goal is to define "real number"!) Another possibility is to begin again with the rational numbers, and to declare real numbers to be certain kinds of sets of rational numbers ("Dedekind cuts") with arithmetic of these sets defined in an appropriate way. Finally, the real numbers may be defined axiomatically as the unique complete Archimedean ordered field; although this last approach does not provide a construction of the real numbers, it gives a list of properties that they must satisfy, which specifies them unambiguously.

It's too long for the lead section. Remember, the purpose of the lead section is to summarize the article; you've used up about all the length the lead section should have, just to define the term. It might go well as an early post-TOC section, called maybe "explanation of the concept" or something. --Trovatore 16:19, 16 May 2006 (UTC)

Looks fine to me, except the third paragraph which I think is too verbose and a bit clumsy. I think the intro could well do without that last paragraph at all. Oleg Alexandrov (talk) 16:59, 16 May 2006 (UTC)

These descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, and an axiomatic definition of the real numbers as the unique completeArchimedeanorderedfield; these definitions are all described in detail below.

I feel the loss of any mention of "gaps" in this version, which has a lot to do with why this discussion got started in the first place, but perhaps that belongs somewhere later on in the article. Dmharvey 17:53, 16 May 2006 (UTC)

I lik the intro you suggest. I think it looks just fine without the gaps, that may be too messy for the intro.

By the way, Gene, if you are reading this, now you see an example of how the community works in there, rather than by pushing one's version through reverts. Oleg Alexandrov (talk) 21:12, 16 May 2006 (UTC)

Oleg, you were the one who started the revert war. The propsals are a nice improvement, except for one thing, they aren't actually in the article. The proposed first paragraph still has the problem that it looks as if it is intended to be a definition. Both "decimal expansion" and "rationals plus irrationals" can be used to define the real numbers, but the number line isn't a definition unless you explain what sort of thing a "line" has to be. It should be clear that corresponding to points on a number line is a property of real numbers, and not an informal definition, unless line is disambiguated.

As for the gaps, it seems to me that giving them the axe and relying only on the LUB property misses something which could be helpful. I had an exposition of what a "gap" was which made perfectly good sense, but which was removed. A place for it makes sense to me. Gene Ward Smith 05:48, 18 May 2006 (UTC)

The propsals are a nice improvement, except for one thing, they aren't actually in the article. That's okay, we have all the time in the world! It will make it there eventually :-)

Let me try to address your concerns. I agree that "decimal expansion" can be used in a rigorous (though slightly painful) definition, I have amended the text accordingly (see below). I'm not so sure about "rationals plus irrationals", because it seems a bit circular -- it sounds like you need a definition of "real number" before you can sensibly say what an "irrational number" is. I have tried using the word "informally" to make it sounds less like a definition to you.

As for the gaps, I have really come around to the view that mentioning rational numbers, irrational numbers, decimal expansions, and the number line, are far more useful to a general audience than any discussion of gaps. I have absolutely no problem with a section elsewhere that expands on this idea, I think it's very important, and I invite you to write it and find a place for it. But I don't think it fits in the introduction.

These descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of "decimal representation"; and an axiomatic definition of the real numbers as the unique completeArchimedeanorderedfield. These definitions are all described in detail below.

Dedekind completeness

So fix the Dedekind cut article. (I think I agree, although I can't really determine the difference at the moment.) — Arthur Rubin | (talk) 21:50, 15 May 2006 (UTC)

How not correct? The LUB property for an ordered group implies that all Cauchy sequences converge, which implies Dedekind closure; and the converse is also true. See Spivak's Calculus. Septentrionalis 17:19, 21 May 2006 (UTC)

Colloquial term

This is not an official term - a look at a general text on economics finds three terms starting with "real" including "real value", but not this one - but is more the sort of colloquialism a reporter might use when discussing a scheduled news item, referred to as a "number" quite often (for example, the regular employment statistics). The phrase "real number" has no particular independent significance, beyond this colloquial use of "number". Elroch 09:08, 16 May 2006 (UTC)

Square root of two or 2?

Why the change? "Square root of two" is usually regarded as better style, though admittedly in a math article it is more common to find numerals such a context. I think the original was fine, and should stand. Gene Ward Smith 05:45, 24 May 2006 (UTC)

"rationals and irrationals"

The current lead says the set of all reals is "the set of all rational numbers and irrational numbers". That makes it sound like a disjunctive kind, which it really isn't. Also, if you follow the irrational number link, it defines those as reals that aren't rational, so it's a bit circular. If this is really the best way to get the concept across in practice, I suppose I can live with it, but I really don't like it. Any alternative suggestions? --Trovatore 20:39, 17 October 2006 (UTC)

Yes, it's bad. At least, the repetition of "number" must go. But unless we want to hit the readers with words like "completion" in the first sentence, I don't see any radical improvement. Septentrionalis 18:59, 18 October 2006 (UTC)

We discussed the lead section at #Introduction a few months ago. I think the text hammered out there is quite okay, so I reverted the edit which changed it radically. -- Jitse Niesen (talk) 01:41, 26 October 2006 (UTC)

"Definition" lacks a definition

The "axiomatic approach" part under the title "Definition" appears a bit strange to me. Why does it first introduce R as the set of real numbers and then state the axioms which are supposed to define real numbers like they were some random properties of R? There is no proper definition of reals under the title "Definition". 130.234.163.82 17:18, 2 March 2007 (UTC)Kelmi

Well, you're right, technically. The definition would be "R is the unique (up to isomorphism) object satisfying the following properties". It's a little awkward to phrase. --Trovatore 19:54, 2 March 2007 (UTC)

Cirticism?

Some sort of acknowledgment of the views of finitists and intuitionists is probably in order, yes. From a brief look, there's nothing obviously unique about Wilderberger's criticisms, and he's most unlikely to be the most notable exponent of them, so I doubt calling him out by name is appropriate. --Trovatore 23:10, 4 June 2007 (UTC)

Missing Section(s)?

there is no 'Axioms' section, despite such bein refrenced in section 'Complete ordered field' Moreover the description of Dedekind (order) completeness now comes after the reference. I browsed the recent article history and it's apparently been vandalized a bunch, perhaps a prior section got nuked. IN any event, perhaps it should just be reinserted/rewritten Zero sharp 04:55, 28 June 2007 (UTC)

Real v Imaginary

Arthur, I kind of think the change you reverted was probably correct. "Real" doesn't make sense as a contradistinction to "complex" -- if that were the case, they'd be called "simple numbers" or some such. I think the idea at the time was, real numbers were, in the natural-language sense, real, whereas imaginary numbers were a figment of the imagination. I suppose this could be folk etymology; we should probably check it out in the historical documents. But it's such a natural explanation that I kind of expect it to be borne out by the facts. --Trovatore 20:30, 2 August 2007 (UTC)

Hmm, yeah, I do think that's probably better, but we really ought to check it out, I suppose. Does anyone have a reference for the etymology? --Trovatore 23:14, 2 August 2007 (UTC)

Trovatore, is pi a complex number? (I know that pi can be expressed as 'pi plus zero times i' , but who thinks of it that way?) CorvetteZ51 12:43, 15 October 2007 (UTC)

Couchy sequence definition

I'm Red75. Current Cauchy sequence definition states that:

A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε provided that n and m are both greater than N.